# Why can't Eisenstein Criterion be used for certain polynomials (to show that it's irreducible over $\mathbb{Q}$)?

Why can't Eisenstein's Criterion be used to show that $$4x^{10} - 9x^{3} + 21x - 18$$ is irreducible over $\mathbb{Q}$?

I mean even if we were to apply Eisenstein here, there doesn't exist a prime $p$ that would apply all the E.C. rules anyways.

A detailed explanation would be great! Thanks.

• @WillJagy but why would you want to do that when you have a degree as big as 10? – dean3794 May 13 '15 at 21:14
• Your cuestión is a bit weird: Eisenstein's criterion cannot be applied to that polynomial simply because there is no prime for which the required condition is satisfied. – Mariano Suárez-Álvarez May 13 '15 at 21:39
• @MarianoSuárez-Alvarez hmm okay. Because working this out as a true and false question o thought it meant that using the Eisenstein Method is not sufficient to show that this polynomial is irreducible – dean3794 May 13 '15 at 21:41
• I'm not sure what your question is. Is it "here is a polynomial that I know to be irreducible, why can't I deduce that it's irreducible from Eisenstein?" because you seemed to have answered that already in the statement of your question - i.e. Eisenstein doesn't apply for any prime $p$, since $3^2 | 18$. – James May 13 '15 at 21:44

## 2 Answers

If $$f(x)$$ is a polynomial in $$\mathbf Z[x]$$ such that $$f(ax+b)$$ is Eisenstein with respect to a prime $$p$$ for some integers (or rational numbers) $$a$$ and $$b$$, then (i) $$f(x)$$ is irreducible over the $$p$$-adic numbers $$\mathbf Q_p$$ and (ii) $$p$$ divides the discriminant of $$f(x)$$. I find with PARI that your polynomial has discriminant with prime factors $$2$$, $$3$$, $$61$$, $$293$$, and $$50997533$$, and for each of these primes $$p$$ I find with PARI that $$f(x)$$ is reducible over $$\mathbf Q_p$$ (it has a factor of degree $$1$$ or $$2$$ in each case). This proves you can't convert $$f(x)$$ into an Eisenstein polynomial by a linear change of variables.

• Hello. Off topic, but I wanted to know if you had a reference for Fermat's own proof of the two square theorem, as you stated here, as a remark on page 28? I couldn't find the entire proof anywhere. – Henry Dec 3 '18 at 16:44
• @Henry there is no record of a detailed proof by Fermat on anything. He usually just claimed to be able to do things, and a few times he very briefly sketched the idea. For the two square theorem, see Remark 5.2 of math.uconn.edu/~kconrad/blurbs/ugradnumthy/descent.pdf. – KCd Dec 3 '18 at 19:58

The only prime that divides 9, 21, and 18 is 3. But $3^2 | 18$, so the Eisenstein criterion does not apply here.

• The polynomial $1+x+x^2+\dots+x^{p-1}$ is irreducible by Eisenestein's theorem. But your method doesn't works. – k1.M May 13 '15 at 21:23
• Eisenstein applies there, but not directly. You must substitute $x\mapsto x+1$. In general, it is not easy to see whether or not this is possible, so I answered the question for the specific polynomial specified by the OP. – William Stagner May 13 '15 at 21:28
• So you can't say that a method doesn't applies, because always there exists tricky methods, and this is mathematics... – k1.M May 13 '15 at 21:30